(461c) Weak Formulation of Supercooled Stefan Problem for Solidification of Pure Silicon in Horizontal Ribbon Growth

Solar energy is a promising source of energy due to its abundance, and has been backed by technological advancements, lower cost and rising global demand. Despite this the silicon wafer costs are still considered high and accounts for 40% of the photo-voltaic supply chain [1]. This is in part because current ingot based wafer production techniques like Czochralski and Bridgman processes---which contribute more than 80% of the silicon substrate for solar cells---are batch processes with low production rates (~2 mm/min) and are highly wasteful [2]. These processes produce up to 55% of material loss while sawing followed by tedious, time consuming grinding and lapping operations. This can be significantly reduced if the wafers could be directly solidified to its correct thickness [3].

Silicon, like water, has a lower density in its solid form than its liquid form. This property of silicon can be used to solidify single-crystal silicon on top of its melt to form a thin sheet of ribbon [4]. The ribbon can then be continuously extracted horizontally by cooling the top surface of the melt. This process, descriptively called Horizontal Ribbon Growth (HRG), has the potential for significant cost savings over traditional wafer production methods [1]. Because the wafers must only be cut to length, there is a substantial decrease in material loss. Additionally in the HRG process, the crystal grows epitaxially in the direction orthogonal to the direction of the heat sink. This makes it possible to produce thin single crystal ribbons at a relatively high speed [5].

Despite its advantages, the HRG process faces fundamental challenges which inhibit its successful operation. One of these is the theoretical modeling of the solidification and heat transfer process around the growing tip of the ribbon. Traditional heat transfer models have been unsuccessful in predicting the upper-limit on the pull speed of the ribbons [6,7]. Finite element simulations of the Stefan problems predict that it is possible to grow a crystal ribbon as thin as desired by increasing the pull speed of the ribbon without limit [9]. However, experimental observations have shown that this is not possible and a very distinct upper bound to the pull speed exists.

Recent numerical models have attributed the pull-speed limit to the non-existence of a stable meniscus or to the formation of a {111} facet at the leading edge [8,9]. To-date no prediction or relationship has been proposed to model the upper bound on pull speed. We identify the reason for this pull speed disparity to be a non-differentiability condition at the tip of the growing ribbon. We conclude that the irregularity of the solid-liquid interface inhibits the use of the classical Stefan formulation to predict the dynamics of the solidifying tip.

To circumvent this problem, we propose to use a weak formulation of the Stefan problem at the solidification boundary. The notion of a sharp interface separating the solid and liquid domains is replaced by a mushy region of a liquid and solid mixture. The phenomenon of supercooling, which is generally observed in the solidification of a pure substance, is modeled using a nucleation dependent phase-temperature rule. This restricts phase change to occur only around a small neighborhood of an existing solid. The proposed model is thus aimed at simulating evolution of a growing solid conditioned on the availability of a nearby nucleating site. The model enjoys the simplicity of the weak formulation, while at the same time incorporating the metastability of the supercooled liquid.

The underlying theory for our heat transfer solution of solidification is reviewed and numerical simulations are presented which highlight the relationship between pull-speed and heat transfer. A finite volume approach was used to study the heat transfer for a variety of boundary conditions including radiation, cooling plate, and helium jets. In each of these cases, an upper limit to the pull speed is observed. The upper limit to the pull speed is shown to be a fundamental consequence of the heat removed from the tip of the ribbon. Recommendations to enhance production speeds and control thickness of the ribbon are also proposed. While our application study focuses on the HRG process, the theory we develop can be applied to other solidification problems as well.

References:

[1] Ranjan, S., Balaji, S., Panella, R. A., & Ydstie, B. E. (2011). Silicon solar cell production. Computers & Chemical Engineering, 35(8), 1439-1453.

[2] Oliveros, G. A., Liu, R., Sridhar, S., & Ydstie, B. E. (2013). Silicon wafers for solar cells by horizontal ribbon growth. Industrial & Engineering Chemistry Research, 52(9), 3239-3246.

[3] William, S. (1962). U.S. Patent No. 3,031,275. Washington, DC: U.S. Patent and Trademark Office.

[4] Eyan P Noronha, German A Oliveros, and Erik B Ydstie. Weierstrass’ variational theory

for the stability of meniscus in ribbon growth processes, under review.

[5] Ke, J., Khair, A. S., & Ydstie, B. E. (2017). The effects of impurity on the stability of Horizontal Ribbon Growth. Journal of Crystal Growth, 480, 34-42.

[6] Zoutendyk, John A. "Theoretical analysis of heat flow in horizontal ribbon growth from a melt." Journal of Applied Physics 49.7 (1978): 3927-3932.

[7] Zoutendyk, John A. "Analysis of forced convection heat flow effects in horizontal ribbon growth from the melt." Journal of Crystal Growth 50.1 (1980): 83-93.

[6] Daggolu, Parthiv, et al. "Thermal-capillary analysis of the horizontal ribbon growth of silicon crystals." Journal of Crystal Growth 355.1 (2012): 129-139.

[8] Helenbrook, Brian T., et al. "Experimental and numerical investigation of the horizontal ribbon growth process." Journal of Crystal Growth 453 (2016): 163-172.